Spherical basis vectors in 3-by-3 matrix form




A = azelaxes(az,el) returns a 3-by-3 matrix containing the components of the basis(e^R,e^az,e^el) at each point on the unit sphere specified by azimuth, az, and elevation, el. The columns of A contain the components of basis vectors in the order of radial, azimuthal and elevation directions.


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Spherical Basis Vectors at (45°,45°)

At the point located at 45° azimuth, 45° elevation, compute the 3-by-3 matrix containing the components of the spherical basis:

A = azelaxes(45,45)
A =

    0.5000   -0.7071   -0.5000
    0.5000    0.7071   -0.5000
    0.7071         0    0.7071

The first column of A is the radial basis vector [0.5000; 0.5000; 0.7071]. The second and third columns are the azimuth and elevation basis vectors, respectively.

Input Arguments

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az — Azimuth anglescalar in range [–180,180]

Azimuth angle specified as a scalar in the closed range [–180,180]. Angle units are in degrees. To define the azimuth angle of a point on a sphere, construct a vector from the origin to the point. The azimuth angle is the angle in the xy-plane from the positive x-axis to the vector's orthogonal projection into the xy-plane. As examples, zero azimuth angle and zero elevation angle specify a point on the x-axis while an azimuth angle of 90° and an elevation angle of zero specify a point on the y-axis.

Example: 45

Data Types: double

el — Elevation anglescalar in range [–90,90]

Elevation angle specified as a scalar in the closed range [–90,90]. Angle units are in degrees. To define the elevation of a point on the sphere, construct a vector from the origin to the point. The elevation angle is the angle from its orthogonal projection into the xy-plane to the vector itself. As examples, zero elevation angle defines the equator of the sphere and ±90° elevation define the north and south poles, respectively.

Example: 30

Data Types: double

Output Arguments

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A — Spherical basis vectors 3-by-3 matrix

Spherical basis vectors returned as a 3-by-3 matrix. The columns contain the unit vectors in the radial, azimuthal, and elevation directions, respectively. Symbolically we can write the matrix as


where each component represents a column vector.

More About

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Spherical basis

The spherical basis vectors (e^R,e^az,e^el) at the point (az,el) can be expressed in terms of the Cartesian unit vectors by


This set of basis vectors can be derived from the local Cartesian basis by two consecutive rotations: first by rotating the Cartesian vectors around the y-axis by the negative elevation angle, -el, followed by a rotation around the z-axis by the azimuth angle, az. Symbolically, we can write


The following figure shows the relationship between the spherical basis and the local Cartesian unit vectors.


MATLAB® computes the matrix A from the equations

A = [cosd(el)*cosd(az), -sind(az), -sind(el)*cosd(az); ...
		cosd(el)*sind(az),  cosd(az), -sind(el)*sind(az); ...
		sind(el),           0,         cosd(el)];
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